Instability thresholds and dynamics of mesa patterns in reaction-diffusion systems
Rebecca McKay and Theodore Kolokolnikov Department of Mathematics and Statistics, Dalhousie University, Canada
Abstract
We consider a class of one-dimensional reaction-diffusion systems,
u = ε2u + f(u, w) t xx . 0= Dw + g(u, w) xx Under some generic conditions on the nonlinearities f,g, in the singular limit ε 0, and for a fixed D independent of ε, such a system exhibits a steady state consisting→ of sharp back-to-back interfaces which is stable in time. On the other hand, it is also known that in the so-called shadow limit D = , the periodic pattern having more than one interface is unstable. In this paper, we analyse∞ in detail the transition between the stable patterns when D = O(1) and the shadow system when D = . We show that this transition occurs when D is exponentially large in ε and we derive instability∞ thresholds D D D . . . such 1 2 3 that a periodic pattern with 2K interfaces is stable if D
1 Introduction
One of the most prevalent phenomena observed in reaction-diffusion systems is the formation of mesa patterns. Such patterns consist of a sequence of highly localized interfaces (or kinks) that are separated in space by regions where the solution is nearly constant. These patterns has been studied intensively for the last three decades and by now an extensive literature exists on this topic. We refer for example to [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein. In this paper we are concerned with the following class of reaction-diffusion models: u = ε2u + f(u, w) t xx (1) 0= Dw + g(u, w) xx in the limit ε 1 and D 1. (2) Under certain general conditions on the nonlinearities f and g that will be specified below, the system (1) admits an interface layer solution. Such a solution has the property that u is either u+ or u− for some constants u = u− everywhere except near the interface location, where it has a layer of size O(ε) + 6 connecting the two constant states u±. A single mesa (or a box) solution consists of two interfaces, one connecting u− to u+ and another connecting u+ back to u−. By mirror reflection, a single mesa can be extended to a symmetric K mesa solution, consisting of K mesas or 2K interfaces (see Figure 1b). The general system (1) has been thoroughly studied by many authors. In particular, the regime D = O(1) is well understood. Under certain general conditions on g and f, it is known that a K-mesa
1 pattern is stable for all K – see for example [1], [11]. On the other hand, in the limit D = , the system (1) reduces to the so-called shadow system, ∞
2 ut = ε uxx + f(u, w0); g(u, w0)=0. (3) Z Equation (3) also includes many models of phase separation such as Allen-Cahn [4] as a special case. Under the same general conditions on g and f, a single interface of the shadow system is also stable; however a pattern consisting of more than one interface is known to be unstable [12]. The main question that we address in this paper is how the transition from the stable regime D = O(1) to the unstable shadow regime D = takes place. In a nutshell, when D is not too large, the stability of K mesa pattern as shown in [1]∞ is due to the stabilizing effect of the global variable w. However as D is increased, the stabilizing effect of w is decreased, and eventually the pattern loses its stability. We find that the onset of instability is caused by the interaction between the interfaces of u. This interaction is exponentially small, but becomes important as D becomes exponentially large. By analysing the contributions to stability of both w and u, we compute an explicit sequence of threshold values D1 >D2 >D3 >... such that a K mesa solution on the domain of a fixed size 2R is stable if D
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0 −1 0 1 2 3 4 5 6 7 (a) (b)
Figure 1: (a) Coarsening process in Lengyel-Epstein model (8), starting with random initial conditions. Time evolution of u is shown; note the logarithmic time scale. Initial inhomogenuities are amplified due to Turing instability. Shortly after, localized structures are formed at around t = 10. This is followed by a long transient pattern consisting of three mesas. The middle mesa eventually disappears around t 100 due to the very slow instability of the three-mesa pattern. The remaining two mesas then move towards∼ symmetrical configuration which appears to be stable. The parameter values are ε = 0.06, a = 10, D = 500, τ = 0.1, domain size is 8. (b) The snapshot of u at t = 5, 000, 000, by which time the system has reached a steady state consisting of a two-mesa pattern.
There are many systems that fall in the class described in this paper. Let us now mention some of them. Cubic model: One of the simplest systems is a cubic model, • u = ε2u +2u 2u3 + w t xx . (4) 0= Dw −u + β xx − 0
2 It is a variation on FitzHugh-Nagumo model used in [5] and in [13]. It is a convenient model for testing our asymptotic results. Model of Belousov-Zhabotinskii reaction in water-in-oil microemulsion: This model is a • was introduced in [14], [15]. In [16], the following simplified model was analysed in two dimensions:
u q u = ε2u f − + wu u2 t xx − 0 u + q − . (5) 0= Dw +1 uw xx − Brusselator model. In [17] the authors considered coarsening phenomenon in the Brusselator • model; a self-replication phenomenon was studied in [11]. After a change of variables the Brusselator may written as u = ε2u u + uw u3 t xx . (6) 0= Dw − β u +1− xx − 0 Gierer-Meinhardt model with saturation. This model was introduced in [18], (see also [19], • [20]) to model stripe patterns on animal skins. After some rescaling, it is
u2 u = ε2u u + t xx − w(1 + κu2) . (7) 0= Dw w + u2 xx − This model was also studied in [21], where stripe instability thresholds were computed. Lengyel-Epstein model. This model was introduced in [22], see also [23]: • 4uv u = u + a u t xx 1+ u2 − − uv . (8) τv = Dv + u t xx − 1+ u2 Other models include: models for co-existence of competing species [24]; vegetation patterns in • dry regions [25]; models of chemotaxis [26] and models of phase separation in diblock copolymers [27].
Figure 1(a) illustrates the instability phenomenon studied in this paper, as observed numerically in the Lengyel-Epstein model. Starting with random initial conditions, Turing instability leads to a formation of a three-mesa pattern at t 10. However such pattern is unstable, even though this only becomes apparent much later (at t ∼100). The resulting two-mesa pattern then drifts towards a symmetric position where it eventually∼ settles. A similar phenomenon for the Belousov-Zhabotinskii model (5) is illustrated in Figure 7(b). It shows the time-evolution a 2-mesa solution to (5) with D>D2, starting with initial conditions that consist of a slightly perturbed two-mesa pattern. After a very long time, one of the mesas absorbs the mass of the other, then moves towards the center of the domain where it remains as a stable pattern. In addition to studying the stability of the mesa patterns, we also study their dynamics. This allows us to describe in detail the mechanism by which the exchange of mass between two mesas can take place, as well as the motion of the interfaces away from the equilibrium. For a pattern consisting of K mesas, we derive a reduced problem, consisting of 2K ODE’s that gouvern the asymptotic motion of the 2K interfaces. The outline of the paper is as follows. In 2 we construct the steady state consisting of K mesas. The construction is summarized in Proposition§ 1. The main result is presented in 3 (Theorem 2), where we analyse the asyptotics of the linearized problem for the periodic pattern, and§ derive the instability thresholds DK . In 4 we derive the reduced equations of motion for the interfaces. In 5 we present numerical computations§ to support our asymptotic results. We conclude with discussion of open§ problems in 6. §
3 2 Preliminaries: construction of the K mesa steady state − We start by constructing the time-independent mesa-type solution to (1). The mesa (or box) solution consists of two back-to-back interfaces. Thus we first consider the conditions for existence of a single interface solution and review its construction. A mesa solution can then be constructed from a single interface by reflecting and doubling the domain size. Similarly, a K-mesa pattern is then constructed by making K copies of a single mesa. We summarize the construction as follows.
Proposition 1 Consider the time-independent steady state of the PDE system (1) satisfying
0= ε2u + f(u, w) xx (9) 0= Dw + g(u, w) xx with Neumann boundary conditions and in the limit
ε 1 and D 1. (10) Suppose that the algebraic system
u+ f(u, w0)du = 0; f(u+, w0)=0= f(u−, w0) (11) Zu− admits a solution u ,u−, w , with u = u−. Define + 0 + 6
g± := g (u±, w0) (12) and suppose in addition that
g− fu (u±, w0) 0; and 0 < < 1. (13) ≤ g− g − + Then a single interface solution, on the interval [0,L] is given by
x l u(x) U − , w w ∼ 0 ε ∼ 0
where U0 is the heteroclinic connection between u+ and u− satisfying
U0yy + f(U0, w0)=0; U u− as y ; U u as y ; (14) 0 → → ∞ 0 → + → −∞ f(U0(0), w0)=0 and l is the location of the interface so that u , 0 4 For future reference, we also define µ± := f (u±, w ) 0; (18) − u 0 ≥ and define constants C± to be such that p − U (y) u− + C−e µ−y, y + ; 0 ∼ → ∞ (19) U (y) u C eµ+y, y . 0 ∼ + − + → −∞ The construction is straightforward and we review it here. First, consider a single interface located at x = l inside the domain [0,L]. We assume that u u for 0 Then U0(y) satisfies the system (14) which is parametrized by w0. In order for such a solution to exist, u± must both be roots of f(u, w0) and U0 must be a hetercolinic orbit connecting u+ and u−. This yields the three algebraic constraints (11) which determine u± and w0. To determine the location l of the interface, we integrate the second equation in (1), and using Neumann boundary condtions we obtain L g(u, w0)dx =0. Z0 Changing variables x = l + εy we estimate 0 (L−l)/ε 0 ε g(U0(y), w0)dy + ε g(U0(y), w0)dy; ∼ − Z l/ε Z0 0 ∞ 0 lg+ + ε [g(U(y), w0) g+] dy + (L l)g− + ε [g(U(y), w0) g−] dy ∼ −∞ − − − Z Z0 Expanding l in ε as in (15) then yields (16) and (17). Since we must have 0 3 Stability of K mesa pattern − We now state the main result of this paper. Theorem 2 Consider the steady state consisting of K mesas on the interval of size 2KL, with Neumann boundary conditions, as constructed in Proposition 1. Suppose that u+ f du fw u− w gw gu > 0 for all x, and > 0. (21) − f g− g u R − + Let 2 3 2 3 2C+µ+ 1 2µ+ 2C−µ− 1 2µ− α+ := u+ exp l ; α− := u+ exp (L l) (22) fwdu ε − ε fwdu ε − ε − u− u− where the constants CR±, µ±,l are as defined in PropositionR 1. Define 2 Lg− D1 := ; (23) 2 (g− g ) α− − + 5 and for K 2, define ≥ 2 Lg− if α+ α− 2 (g− g+) α− Lg− 2 + D := if α− α+ K 2 (g− g+) α+ − −1 2 2 L 1 1 2α+α− (1 cos π/K) g+g− −2 −2 + − 2 if O(α+)= O(α−) 2 (g− g ) g α + g α− 2 4 − 2 2 + + + − s 4 g−α+ + g+α− ! − (24) Then the K mesa pattern is a stable equilibrium of the time-dependent system (1) if D Theorem 2 follows from a detailed study of the linearization about the steady state. We consider small perturbations of the steady state of the form u(x, t)= u(x)+ φ(x)eλt, w(x, t)= w(x)+ ψ(x)eλt where u(x), w(x) denotes the K-mesa equilibrium solution of (9) on the interval of length 2KL with Neumann boundary conditions, whose leading order asymptotic profile was constructed in Proposition 1. For small perturbations φ, ψ we get the following eigenvalue problem, 2 00 λφ = ε φ + fu(u, w)φ + fw(u, w)ψ 00 . (25) 0= Dψ + g (u, w)φ + g (u, w)ψ u w with Neumann boundary conditions. The sign of the real part of the eigenvalue λ determines the stability: the system is said to be unstable if there exists a solution to (25) with Re(λ) > 0; it is stable if Re(λ) < 0 for all solutions λ to (25). To analyse the stability for Neumann boundary conditions, the first step is to consider periodic boundary conditions. The central analysis is summarized in the following lemma. Lemma 3 (Periodic boundary conditions) Consider the steady state consisting of K mesas on the interval of size 2KL, as constructed in Proposition 1, and consider the linearized problem (25) with periodic boundary conditions φ ( L)= φ(2KL L) − − The linearized problem admits 2K eigenvalues. Of these, 2K 2 are given asymptotically by − u+ f du ± − w λ (a b ) u (26) θ ∼ ± | | 0 2 R −L ux where R (g+ g−) L g+l a = α + α− + − (27) + D 1 cos θ − D − 2 2 2 2 (g+ g−) L (α+ + α−) b = α + α− +2α α− cos θ + − lα (L l)α− (28) | | + + D 1 cos θ − + − − 2 − (g g−) + + − L2 2 (1 cos θ) l(L l) D2 (1 cos θ)2 − − − − with 2 3 2 3 2C+µ+ 1 2µ+ 2C−µ− 1 2µ− α+ = u+ exp l ; α− = u+ exp (L l) ; (29) fwdu ε − ε fwdu ε − ε − u− u− and R R θ =2πk/K; k =1 ...K 1. (30) − 6 The other two eigenvalues are λ =0 (for which the corresponding eigenfunction is (φ, ψ) = (u0, w0)) and u+ f du g− g+ u− w λeven − 0 (31) ∼−σ+l + σ−(L l) u2 − R −L x where σ± are given in (41). In the case θ = π, the formula (26)R simplifies to u+ u+ 2 fwdu 2 fwdu − g−L u− + g+L u− λπ = 2α− 0 ; λπ = 2α+ 0 (32) − D(g− g+) u2 − D(g− g+) u2 − R −L x − R −L x Proof of Lemma 3. The idea is toR make use of Floquet theory. That is, instead ofR considering (25) with periodic boundary conditions on [ L, 2KL L], we consider (25) on the interval [ L,L] with the boundary conditions − − − 0 0 0 0 φ(L)= zφ( L), φ (L)= zφ ( L); ψ(L)= zψ( L), ψ (L)= zψ ( L), (33) − − − − We then extend such solution to the interval [L, 3L] by defining φ(x) := zφ(x 2L) for x [L, 3L] and similar for ψ. This extension assures continuity of φ, ψ and φ0, ψ0 at L. Morever,− since∈u, w are periodic with period 2L, it is clear that φ, ψ extended in this way satisfies (25) on [ L, 3L] and moreover φ(3L)= z2φ( L). Repeating this process, we obtain solution of (25) on the whole interval− [ L, 2KL L] with φ(2KL −L)= φ( L)zK . Hence, by choosing − − − − z = exp(2πik/K) , k =0 ...K 1, − we have obtained a periodic solution to (25) on [ L, 2KL L]. To solve (25) subject to (33), we estimate the− eigenfunction− s as φ c±u ; ψ ψ ( l) when x l. (34) ∼ x ∼ ± ∼± Note that 2 0= ε uxxx + fuux + fwwx. Multiplying (25) by u and integrating by parts on [ L, 0] we then obtain x − 0 0 2 2 0 λc− ux ε (φxux φuxx)−L + fw (ψux φwx) − ∼ − − − Z L Z L We note that the integral term on the right hand side is dominated by the contribution from x = l. Using the anzatz (34) we then obtain − 0 u+ 2 2 0 λc− ux ε (φxux φuxx)−L + (ψ( l) c−wx ( l)) fwdu (35) − ∼ − − − − Z L Zu− Similarly on the interval [0,L] we get L u+ λc u2 ε2 (φ u φu )L (ψ(+l) c w ( l)) f du (36) + x ∼ x x − xx 0 − − +. x − w Z0 Zu− Write (35) and (36) as L c+ κ1 ( φuxx)0 ψ(+l)+ c−wx(+l) λκ0 = − 0 − (37) c− κ ( φu )− + ψ( l) c−w ( l) 1 − xx L − − x − ! where 0 2 2 −L ux ε κ0 = ; κ1 = . u+ f du u+ f du uR− w u− w R R 7 We now transform (37) into an matrix eigenvalue problem. To do so , we will express the boundary terms as well as ψ ( l) in terms of c±. Determining± ψ ( l) . We start by estimating ± −l+ u+ guuxdx gudu g+ g−; − − ∼ ∼ − Z l Zu− +l+ u− guuxdx gudu g− g+ − ∼ ∼ − Z+l Zu+ ±l+ ± ± where ±l− denotes integration over any interval that includes l and g = g (u , w0) . On the other hand, φ is dominated by the contribution from the interfaces. Hence± we estimate R g φ c− (g g−) δ (x + l)+ c (g− g ) δ (x l) (38) u ∼ + − + − + − where δ is the delta function. Therefore we write (g+ g−) ψ (x) − (c−η(x; l) c η(x; l)) ∼− D − − + where η(x; x0) is a Green’s function which satisfies σ(x) η00 + η = δ (x x ) (39) D − 0 with boundary conditions η (L)= zη( L), η0(L)= zη0( L), z = exp(2πik/2K) , k =0 ...K 1 (40) − − − where σ+, x As will become apparent, the case z = 1 is special and will be considered later. For now, assume z =1. Then to leading order, η must satisfiy 6 − − η = 0; η x ; x = η x+; x ; η0 x+; x η0 x ; x =1. xx 0 0 0 0 0 0 − 0 0 so that A + (x + L)B, x A +2BL + L x = zA; 1+ B = zB − 0 We then obtain z 1 2L + (L x ) (z 1) B = − ; A = − 0 − (z 1)2 (z 1)2 − − 2Lz η(l; l)= η( l; l)= ; (42) − − (z 1)2 − 2Lz +2zl(z 1) η(l; l)= − ; (43) − (z 1)2 − 2Lz +2l(1 z) η( l; l)= − = η(l; l). − (z 1)2 − − 8 In summary, we obtain ψ (l) (g+ g−) η(l; l) η(l; l) c − − − + (44) ψ ( l) ∼ D η(l; l) η(l; l) c− − − − − where η(l; l), η(l; l) are given by (42) and (43). Boundary terms.− Next we compute the boundary terms. We start by estimating the behaviour of u and φ near L. Since u0( L)=0, we have x − − x + L u u− + A [exp(µ−z) + exp( µ−z)] , z = . (45) ∼ − ε The constant A is found by matching u to the heteroclinic solution as x : U (y) u− + C− exp(µ−y) ; ∼ x + l x + l u(x) U( ) u− + C− exp µ− . (46) ∼ ε ∼ ε Matching (45) and (46) we then obtain µ− A = C− exp (L l) . − ε − Performing a similar analysis at x =0 and at x = L we get: 2 2 00 µ− µ− 00 µ+ µ+ u ( L)=2C− exp (L l) ; u (0) = 2C exp l . ± ε2 − ε − − + ε2 − ε Next we estimate φ( L). Near x L we write − ∼− µ− µ φ = C exp (x + L) + C exp + (x + L) 1 ε 2 − ε 0 where C1 and C2 are to be determined. Away from L, we have φ c−u . Matching the decay modes, we then obtain − ∼ µ− µ− C = c−C− exp (L l) . 1 ε − ε − On the other hand, near x +L we write ∼ µ− µ φ = C exp (x + L) + C exp + (x + L) ; 3 ε 4 − ε as before, we get µ µ C = c C + exp + (L l) . 4 − + + ε − ε − The constants C2 and C3 are determined by using the boundary condtions (40), which yields C3 = zC1; C4 = zC2 In summary, we get µ− µ− 1 φ( L) C− exp (L l) c− c ; − ∼ ε − ε − − z + µ− µ− φ(L) C− exp (L l) [zc− c ] . ∼ ε − ε − − + Performing a similar analysis at x 0, we obtain ∼ µ+ µ+ φ(0) C exp l [c− c ] . ∼ + ε − ε − + 9 We thus obtain 3 3 L 2 µ− 2µ− 2 µ+ 2µ+ (φu ) =2C− exp (L l) [zc− c ]+2C exp l [c− c ] ; xx 0 ε3 − ε − − + + ε3 − ε − + 3 3 0 2 µ− 2µ− 1 2 µ+ 2µ+ (φu ) = 2C− exp (L l) c− c 2C exp l [c− c ] ; xx −L − ε3 − ε − − z + − + ε3 − ε − + so that L κ1 (φxux φuxx)0 α+ + α− α+ zα− c+ − 0 = 1 − − (47) κ (φ u φu ) α+ α− α+ + α− c− " 1 x x − xx −L # − − z where α± are given by (22). Finally, we estimate g l w0(l) + w0( l). (48) ∼− D ∼− − Substituting (47), (48) and (44) into (37) we obtain c+ a b c+ λκ0 = c− ¯b a c− where (g− g+) g+l (g− g+) a = α + α− + − η(l; l) ; b = α zα− − η(l; l). + D − D − + − − D − It follows that λκ = a b . 0 ± | | Next we compute 2 (g− g+) Lz g+l a = α+ + α− + − D (z 1)2 − D − 2 (g− g+) Lz + zl(z 1) b = α+ zα− − − − − − D (z 1)2 − 1 2 (g− g+) Lz l(z 1) ¯b = α+ α− − − − − − z − D (z 1)2 − 2 2 2 2 2 (g− g+) 2Lz 2 (g− g+) L z +1 − − b = α+ + α− + α+α (z +¯z)+ α+ − 2 + l + α − 2 l | | D (z 1) ! D (z 1) − ! − − 2 2 2 4 (g− g+) z L zl(L l) + −2 4 + − 2 D (z 1) (z 1) ! − − We write z = eiθ, φ =2πk/K and note that 2z 1 z2 +1 cos θ = ; = . (z 1)2 cos θ 1 (z 1)2 cos θ 1 − − − − Combining these computations, we obtain (26), (27), (28), provided that z =1. Next we consider (33) with z = 1, which corresponds to periodic boundary6 conditions on [ L,L]. − This admits two solutions. One is λ = 0 corresponding the odd eigenfunction φ = ux, ψ = wx. The other eigenfunction is even. This corresponds to imposing the boundary conditions φ0(0)=0= φ0(L); ψ0 (0)=0= ψ0 (L) . As before, we assume φ u ; ψ ψ (l) when x l. (49) ∼ x ∼ ∼ 10 and obtain L u+ λ u2 ε2 (φ u φu )L (ψ(l) w (l)) f du (50) x ∼ x x − xx 0 − − x w Z0 Zu− As before, we obtain (g g−) ψ (x) + − η(x; l) ∼ D 0 0 where η(x; x0) satisfies (39) with boundary conditions η (0)=0= η (L). We then obtain 1 1 η L . ∼ σ(x) ∼ σ+l + σ−(L l) 0 − The boundary term is evaluated as previously,R but is of smaller order. This yields the formula (31) for the even eigenvalue. We now use Lemma 3 to characterize stability with Neumann boundary conditions as follows. Suppose that φ has Neumann boundary conditions on [0,a]. Then we may extend φ by even reflection around the origin; it then becomes periodic on [ a,a]. From this principle, it follows that the eigenvalues of a K mesa steady state with Neumann boundary− conditions form a subset of the eigenvalues of 2K spikes with periodic boundary conditions. On the other hand, if φ is an eigenfunction on [ a,a] with periodic − boundary conditions then so is φ( x) and hence φˆ(x)= φ(x)+ φ( x) is an eigenfunction on [0,a] with − 0 − Neumann boundary conditions, provided that φˆ(x) =0. Since φˆ (0) = 0 and φ satisfies a 2nd order ODE, 6 φˆ = 0 iff φˆ(0) = 0 iff φ(0) =0. Verifying this condition, we obtain the following result. 6 6 6 Lemma 4 (Neumann boundary conditions) Consider the steady state consisting of K mesas on the interval of size 2KL, with Neumann boundary conditions. The linearized problem admits 2K eigenvalues. Of these, 2K 2 are given asymptotically by (26) to (22) of Lemma 3, but with − θ = πk/K, k =1 ...K 1. (51) − The additional two eigenvalues correspond to an even and odd eigenfunction with Neumann boundary condtions on [ L, +L]. They are − u 2 + f du g−L u− w λodd = 2α− 0 (52) − D(g− g+) u2 − R −L x u+ f du g− g+ u− wR λeven = − 0 (53) −σ+l + σ−(L l) u2 − R −L x with all the symbols as defined in Lemma 3. R Figure 2 shows the actual numerical computation of the four distinct eigenvalues/eigenfunctions for the cubic model (4) with K = 2. (see 5 for methods used). Note that φ is localized at the interfaces and is nearly constant elsewhere; whereas§ ψ has a global variation. An excellent agreement between the asymptotic results and numerical computations is observed. ± Critical thresholds. To obtain instability thresholds, we set λθ = 0 in Lemma 3; we then obtain 2 g−L a b = 0. Using l = − and after some algebra we obtain: − | | g− g+ 2 2 2 2 2 g−α+ + g+α− 2 g+g− − 0=2α+α (1 cos θ) D 2L D + L 2 (54) − − g− g+ (g− g ) − − + + which implies that λθ = 0 iff D>Dθ where 2 Lg+ if α+ α− 2 (g− g+) α− Dθ − 2 ∼ Lg− if α− α+ 2 (g− g ) α − + + 11 u w 1 0.003 0.5 0 –0.5 0 –0.001 –1 –1 0 1 2 3 –1 0 1 2 3 φ ψ λ 1 –0.8565 0.5 –0.857 0 –0.8575 –0.3 λeven = 1.557 –1 0 1 2 3 –1 0 1 2 3 asympt =− 1.560 − 1 0.002 0.5 0 0 –0.5 –0.002 –1 λodd = 0.001192 –1 0 1 2 3 –1 0 1 2 3 asympt =− 0.001194 1 0.01 − 0.5 0 0 –0.5 − –1 –0.01 λ = 0.01275 –1 0 1 2 3 –1 0 1 2 3 π/2 − asympt = 0.01285 1 − 0.5 0.001 0 0 –0.5 –0.001 –1 λ+ = 0.0001936 –1 0 1 2 3 –1 0 1 2 3 π/2 − asympt = 0.0001950 − Figure 2: Top row: steady-state with two mesas. The cubic model (4) was used with L =2,ε =0.13,D = 40,β0 = 0.3. Bottom four rows: the four possible eigenfunctions and the corresponding eigenvalues. The numerical− computations are described in 5 Asymptotic value is computed using (26). Excellent agreement is observed in all cases (less than 1%§ error). 12 and more generally, without any assumptions on α− and α+, −1 2 2 L 1 1 2α+α− (1 cos θ) g+g− Dθ = − − + − . 2 2 2 2 2 2 (g− g+) g− α− + g+ α+ 2 4 − 4 g α + g α− − s − + + ! ± It is clear that Dθ is an increasing function of θ; it is also easy to verify that λθ < 0 if α± is decreased sufficiently, or equivalently, if D is sufficiently small: in this case the formula (26) reduces to ± (g g−) L ld g l λ κ + − 1 1 2 (1 cos θ) + . (55) θ 0 ∼ D2 (1 cos θ) ± − − L2 − D − s ! On the other hand, when K = 1, the eigenvalues are λodd and λeven, given by (52, 53). It is clear that λeven < 0 for all D; on the other hand setting λodd = 0 yields the threshold (23). This completes the proof of Theorem 2. 4 Dynamics We now derive the equations of motion of quasi-stable fronts. This will allow us to describe the dynamics of the fronts that are not necessary in a symmetric pattern. In addition, this will also enable us to describe in more detail the aftermath of an instability of a symmetric pattern. We assume that the pattern consists of K mesas on the interval of length 2KL. Each mesa is bounded by two interfaces located at xli and xri and we assume the ordering L < x < x < x < x < < x < x < (2K 1)L. − l1 r1 l2 r2 · · · lK rK − Moreover to leading order we assume u , if x (x , x ) for some i (1,K) u + ∈ ri ri ∈ ∼ u−, otherwise and near each interface, u (x + εy) U( y), u (x + εy) U(y), y = O(1), i =1 ...K, li ∼ − ri ∼ where U is the heteroclinic orbit given in (14), with U(y) u± as y . We also suppose that xli, xri are slowly changing with time. In addition we define: → → ∓∞ x + x x := li ri , i =1 ...K; ci 2 xri + x x := l(i+1) , i =1 ...K 1; x := L, x := (2K 1)L. di 2 − d0 − dK − The equations of motions are derived from 2K solvability conditions about each interface. First consider the interface xl1. We expand 1 1 u(x, t)= u (z)+ u , w(x, t)= w + w 0 D 1 0 D 1 where w0 is given by (11) and z = x x (t); u (z)= U ( z/ε) . − l1 0 − 1 Expanding in terms of D we obtain 0= εu0zz + f(u0, w0); (56) x0 (t)Du0 = ε2u + f (u , w )u + f (u , w )w ; (57) − l1 0 1zz u 0 0 1 w 0 0 1 0= w1xx + g(w0,u0). (58) 13 0 1 0 We will see later that xl1 = O( D ) so the above expansion is indeed consistent. We multiply (57) by u0 and integrate on x ( L, x ). Upon integrating by parts we obtain: ∈ − c1 xc1 xc1 0 2 2 x=xc1 xl1(t)D (u0x) dx ε (u1zu0z u1u0zz)x=−L + fww1. − − ∼ − − Z L Z L The boundary term is evaluated similarly as in Section 3. The end-result is, 2 x=xc1 2 µ+ 2 2µ− ε (u u u u ) =2D C µ exp (x x ) + C−µ− exp (L + x ) . 1z 0z − 1 0zz x=−L − + + − ε r1 − l1 − ε l1 The integral terms are estimated as xc1 ∞ 2 xc1 u+ 2 1 dU (u0x) dx dy; fww1 w1(xl1) fw(w0,u)du. − ∼ ε −∞ dy − ∼ Z L Z Z L Zu− Similar analysis is performed at each of the remaining interfaces. In this way, we obtain the following system: ε x0 (t) (BT ) 1 w (x ) u+ f (w u)du li 2 li D 1 li u− w 0, ∼ ∞ dU − −∞ dy R , i =1 ...K (59) 0 ε u+ x (t) (BT ) + 1 w (x ) f (w u)du ri R 2 ri D 1 ri u− w 0, ∼ ∞ dU −∞ dy R R where 2 2µ− 2 µ+ (BT ) = 2C−µ− exp (L + x ) +2C µ exp (x x ) (60) l1 − − ε l1 + + − ε r1 − l1 2 µ− 2 µ+ (BT ) = 2C−µ− exp x x − +2C µ exp (x x ) , i =2 ...K 1 li − − ε l1 − r(i 1) + + − ε ri − li − (61) 2 µ− 2 µ+ (BT ) = 2C−µ− exp (x x ) +2C µ exp x x , i =2 ...K 1 ri − − ε ri − li + + − ε l(i+1) − ri − (62) 2 µ− 2 2µ+ (BT ) = 2C−µ− exp (x x ) +2C µ exp ((2K 1)L x ) , (63) rK − − ε rK − lK + + − ε − − rK The constants w1(xli) and w1(xri) are obtained by recursively solving for w1 which satisfies: 00 g , x [x , x ], i =1 ...K 0 0 w = + ∈ li ri ; w ( L)=0= w ((2K 1)L). 1 g− otherwise 1 − 1 − To simplify the expression for w1, we first define the interdistances xl1 + L, i =0 m = x x , i =1 ...K 1 ; p = x x , i =1...K. i l(i+1) ri i ri li (2K − 1)L x , i = K− − − − ri We obtain the following recursion formulae, 0 g−m0, i =1 w (xli)= 0 − w (x − ) g−m , i =2 ...K r(i 1) − i w0(x )= w0(x ) g p , i =1 ...K; ri li − + i m2 − 0 w( L) g 2 , i =1 w (xli)= − − 2 0 mi ( w xr(i−1) + w (xr(i−1))mi g− , i =2 ...K − 2 2 0 p w(x )= w(x )+ w (x )p g i , i =1 ...K. ri li li i − + 2 14 Expanding, we obtain 2 m0 w (x )= w( L) g− 1 l1 − − 2 2 2 m0 p1 w (x )= w( L) g− + m p g 1 r1 − − 2 0 1 − + 2 2 2 2 m0 m1 p1 w (x )= w( L) g− + m p + m m + g + p m 1 l2 − − 2 0 1 0 1 2 − + 2 1 1 2 2 2 2 m0 m1 p1 p2 w (x )= w( L) g− + m p + m m + + m p + m p g + p m + p p + 1 r2 − − 2 0 1 0 1 2 0 2 1 2 − + 2 1 1 1 2 2 ... The general pattern is 2 i−1 i−1 i−1 i−1 i−1 mj w(x )= w ( L) g− m m + m p + li 1 − − j=0 k=j+1 j k j=0 k=j+1 j k j=0 2 − − − − 2 (64) i 1 i i 1 i 1 i 1 pj g+ P Ppj pk + P pjmPk + P − j=1 k=j+1 j=1 k=j j=1 2 2 i−1 i−1 i−1 i i−1 mj w(x )= w ( L)Pg− P mP m P+ P m p + ri 1 − − j=0 k=j+1 j k j=0 k=j+1 j k j=0 2 − 2 . (65) i i i i 1 i pj g+ P Ppj pk + P pj mPk + P − j=1 k=j+1 j=1 k=j j=1 2 P P P P P It remains to determine the constant w1( L); this is done by considering the conservation of mass as follows. Integrating the equation for w in (1)− we obtain that for all time t, g− mj + g+ pj = 0; X X moreover m =2KL p so that j − j P P 2KLg− (xri xli)= . (66) − g− g+ X − Differentiating (66) with respect to t and substituting into (59) we then obtain, K 1 u+ (BT ) (BT ) + [w (x )+ w (x )] f (w u)du =0. (67) ri − li D 1 ri 1 li w 0, i=1 u− X Z Substituting (60-63), and (64-65) into (67) then determines the constant w1( L). xl1+xr1− Dynamics of a single mesa. For a single mesa, we define x0 = 2 to be the midpoint of the mesa. Due to mass conservation, we have g− xl1 = x0 l, xr1 = x0 + l; l = L. (68) − g− g − + 0 0 0 Substituting (68) and x0 = (xl1 + xr1)/2 into (59) and after some algebra we then obtain, 2 2µ− 2 2µ+ 2C−µ− exp ε (L l + x0) +2C+µ+ exp ε (L l x0) d ε − − − − u+ − − x0 = 2 (69) 2 g− − fw(w0,u)du dt ∞ dU 2 u 2 x0 −∞ dy − D g− g R − + R Note that for the special case where C± = C0; µ± = µ0 the formula further simplifies to 2 u+ g− fw(w0,u)du d ε 2 2µ0 2µ0 2 u− x0 = 2 C0µ0 exp (L l) sinh (x0) x0 . (70) dt ∞ dU − ε − ε − D R g− g+ ! 2 −∞ dy − R 15 5 Numerical computations In this section we validate our asymptotic results by a direct numerical computations of the full system (1), as well as the linearized equations (25). Let us first describe the numerical methods used. To perform the numerical simulation of the full system (1) we used the standard software FlexPDE [28]. It uses a FEM-based approach and automatic adaptive meshing with variable time stepping. We used a global error tolerance of errtol=0.0001 which is more than sufficient to accurately capture the interface dynamics [we also verified that changing the error tolerance did not change the solution]. To determine the solution to the linear problem (25), we have reformulated it as a boundary value dλ problem by adjoining an extra equation dx = 0 as well as an extra boundary condition such as ψ( L)= 1. We used the asymptotic solution derived in 2 as our initial guess. Maple’s dsolve/numeric/bvp− routine was then used to solve the resulting boundary§ value problem (this routine is based on a Newton- type method, see for example [29]). Unfortunately, we found that sometimes the Newton’s method failed to converge, especially for problems with several interfaces. So we resorted to second method: we discretized the laplacian using the standard finite differences, thus converting (25) to a linear algebra matrix eigenvalue problem. On the other hand, this second approach is much less accurate; especially since the required eigenvalue is very small. Because of this, we used the combination of the two approaches: we used method 2 as initial guess to the boundary value problem solver. This finally converged with sufficient precision. 5.1 Cubic model We now specialize our results to the cubic model (4), f = 2(u u3)+ w; g = β u. − 0 − Let us first consider a symmetric mesa solution on interval [ L,L], with its maximum at zero. For such a solution, we find − w = 0; u− = 1, u = +1; U(y)= tanh (y) ; (71) 0 − + − g = β 1, g− = β +1; (72) + 0 − 0 ∞ u+ 2 4 Uy = ; fw =2; (73) −∞ 3 Z Zu− β +1 l = 0 L; l =0; (74) 0 2 1 1 2 µ± = 2; C± = 2; α± = 32 exp (1 β) L . (75) ε − ε ± One of the advantages of using the cubic model as a test case is that due to symmetry, l1 =0. This means that the asymptotic results are expected to be very accurate. We obtain the following expressions for λodd and λeven : λ 12ε (76) even ∼− 3 (β + 1)2 Lε 2L λ 0 + 96exp (1 β ) (77) odd ∼− 4 D − ε − 0 The even eigenvalue λeven is always stable; the odd eigenvalue λodd becomes unstable as D is increased past the critical threshold D1 is given by (β + 1)2 Lε 2L D = 0 exp (1 β ) . (78) 1 128 ε − 0 with λodd < 0 when D 16 2000 1800 200 1600 1400 150 1200 1000 100 800 600 400 50 200 0 –0.8 –0.4 0 0.20.40.60.8 1 0 –0.8 –0.4 0 0.20.40.60.8 1 (a) (b) 10000 10000 8000 8000 6000 6000 4000 4000 2000 2000 1 2 3 4 1 2 3 4 (c) (d) Figure 3: (a) Dynamics of a single mesa for the cubic model (4) with with β0 = 0.2; ε = 0.22, D = 20,L =1. Vertical axis is time, horizontal axis is space. The contour u = 0 is shown.− Solid lines are the asymptotic results derived in 4. Dots represent the output of the full numerical simulation of (4) using § FlexPDE. The initial conditions are given by (83) with x0 = 0.15. The solution moves to the left and converges to a symmetric one-mesa pattern. (b) Same as in (a), but x0 =0.16. The solution moves to the right until it merges with the boundary. (c) Dynamics of two-mesas. The parameters are K =2,L = 1(so that x [0, 4]); β0 = 0.3, ε =0.13, D = 70 Initial interface locations are 0.8, 1.5, 2.3, 3.0. The solution converges∈ to the symmetric− two-mesa solution. (d) Same as (c) except that D = 85. The two-mesa solution is unstable; the right mesa absorbs the left, eventually resulting in a stable one-mesa pattern which slowly moves to the center. and the equations of motion for a single mesa become dx 3 (β + 1)2 1 ε 4x 1 0 = 0 Lε sinh 0 x , (79) dt 4 D 4 ε − D 0 1 xl1+xr1 where x0 = 2 is the center of the mesa. Note that ∂ dx0 = λodd ∂x0 dt x0=0 so that the linearization of the equations of motion around the symmetric equilibrium agrees with the full linearization of the original PDE; the equilibrium x0 = 0 undergoes a pitchfork bifurcation and becomes unstable as D increases past D1. For K symmetric mesas on the interval of length 2R, we have L = R/K 17 and the thresholds (24) become: 2 R (1 β0) ε − K exp 2R (1 + β ) , if β < 0; 128 εK 0 0 DK 2 ; K 2 (80) ∼ (1 + β0) Lε 2R ≥ exp (1 β0) , if β0 > 0 128 εK − Finally, if we take the “inverted” mesa with u +1 near the boundaries, by changing the variables ∼ u u, w w, the model remains the same except β0 is replaced by β0. Thus the stability threholds→ − for the→ −inverted mesa are − (1 β )2 Lε 2L Di = − 0 exp (1 + β ) (81) 1 128 ε 0 2 R (1 β0) K ε 2R − exp (1 + β0) , if β0 < 0; i 128 εK DK 2 R ; K 2 (82) ∼ (1 + β0) K ε 2R ≥ exp (1 β0) , if β0 > 0 128 εK − We now numerically validate our asymptotic results by direct comparison with full numerical simula- tion of the system (4). Mesa dynamics: single mesa. Choose L = 1,ε = 0.22 and β = 0.2. From (78) we then get 0 − Dc = 60.138. Now suppose that D = 20. Then the ODE (79) admits three equilibria: x0 = 0 (stable) and x± = 0.156 (both unstable). We now solve the full system. Take initial conditions to be ± (x x )+ l (x x ) l u(x, 0) = tanh − 0 tanh − 0 − 1; w(x, 0)=0. (83) ε − ε − This corresponds to a mesa solution of length l centered at x . Thus if x ( 0.156, 0.156) then we 0 0 ∈ − expect the mesa to move to the center and stabilize there. On the other hand, if x0 > 0.156 then the mesa will move to the right until it merges with the right boundary. In Figure 3, we plot the numerical simulations for x0 =0.150 and x0 =0.160. The observed behaviour agrees with the above predictions. Dynamics of two-mesa solution. Here we consider a two-mesa solution. We take the domain x [0, 4] (i.e. L = 1,K = 2) and take β0 = 0.3, ε = 0.13. From (74), we get l = 0.35 so that the symmetric∈ equilibrium location of the interfaces− are 1 0.35 and 3 0.35 which yields 0.65, 1.35, 2.65, 3.35. According to (80), the two-mesa symmetric configuration± is stable± provided that D< 82, and is unstable otherwise. To verify this, we solve the full system with initial interface locations given by 0.8, 1.5, 2.3, 3.0. These are relatively close to the symmetric equilibrium. We found that when D< 80, such configuration converges to the symmetric two-mesa equilibrium; however it is unstable if D > 80 – see Figure 3(c,d). This is in good agreement with the the theoretical threshold D2 = 82. Next we also compute the four eigenvalues for several values of D, and compare to asymptotic results, + shown in Figure 4. An excellent agreement is once again observed, including the crossing of zero for λπ/2 at D = 82. The transitional case of β0 = 0. This is the degenerate case for which the formula (80) does not apply. In this case, α+ = α− and the formula (24) reduces to L D ε exp(2L/ε); β =0,K 1. (84) K 2 π 0 ∼ 256 cos 4K ≥ (this formula is also valid when K = 1, as can be verified by comparing it to (78)). Note that this is also qualitatively different from β = 0, in that D actually depends on K when β = 0. To validate 0 6 K 0 (84) numerically, we set ε = 0.17, L = 1,β0 = 0. Formula (84) then yields the asymptotic thresholds π/K D1 = 170.8, D2 = 100.1, D3 = 91. Next, we have computed the eigenvalues λ+ explicitly using the full formulation (25) for K =1, 2, 3 several different D and for ε,L as above; these are shown in Figure 5. An excellent agreement can be observed with the predicted threshold values. For example for D = 95, a two-mesa solution on the interval of size 4 is stable but a three-mesa on the interval of size 6 solution is not. 18 − + λeven λ λodd λπ/2 π/2 –1.555 –0.0005 –0.005 0 –1.556 –0.001 –0.0001 –0.01 –1.557 –0.0002 –0.015 –0.0015 –1.558 –0.0003 –0.02 –1.559 –0.002 –0.0004 –0.0005 –0.025 –1.56 20 40 60 80 100 120 20 40 60 80 100 120 40 60 80 100 120 20 40 60 80 100 120 Figure 4: The four eigenvalues of the two-mesa pattern of (4) as a function of D. Other parameters are as in Figure 3(c,d). Circles represent numerical computations of (25); lines are the asymptotic results + given by (26). Excellent agreement is observed, including the crossing of λπ/2 at D = D2 = 82. 140 10000 D 120 0.0002 λ 1 100 1000 80 0.0001 λ+ π/3 100 60 60 80 100 120 140 160 180 200 220 240 D2 –0.1 0 0.1 0 10 D D3 –0.0001 + 1 λπ/2 –0.0002 λodd 0.1 –0.0003 –1 –0.5 0 0.5 1 β0 (a) (b) Figure 5: Instability thresholds of the K-mesa pattern in the cubic model with K = 1, 2 and 3. (a) L = 1,ε = 0.17,β0 = 0. Circles show λ as computed by numerically solving the full formulation (25) for different values of D and the three different modes, as indicated. Solid curves are the asymptotic approximations for λ as given by (26). The K-mesa pattern is unstable for D >DK where D1 = 171, D2 = 100, D3 = 91. (b) The graph of DK versus β0 with L = 1,ε = 0.17, as given by Theorem 2 (note the logarithmic scale). The insert shows the zoom near β0 =0. Boundary-mesa versus interior mesas. Let us now compare the stability properties of interior mesas versus patterns with half-mesas attached to the boundary. The latter are equivalent to an “inverted mesa” patterns. This situation is shown in the Figure 6. Fix ε =0.15,L =1. Moreover β = 0.1 < 0 so that the roof of the mesa occupies more space than 0 − its floor (l =0.45 < 1/2). In this case, the instability threshold for a single mesa is (78), D1 2223 and i ∼ for the inverted mesa it is (81), D1 = 230. Moreover the instability thresholds for K interior mesas on the interval 2LK with K > 1 is also (80) DK 230. This threshold is also the same for two boundary mesas or K inverted mesas (82). ∼ 5.2 Model of Belousov-Zhabotinskii reaction in water-in-oil microemulsion The cubic model is unusual in the sense that due to the symmetry of the interface, the correction to interface length l1 of Proposition 1 is zero. To see the more usual case when it is not – and a dramatic effect that it can have on the accuracy of asymptotics – consider the Belousov-Zhabotinskii model (5): 19 D1 = 2223.4 D2 = 230.78 1 1 0 0 –1 –1 –1 0 1 –1 0 1 2 3 (a) (b) i i D1 = 230.78 D2 = 230.78 1 1 0 0 –1 –1 0 1 2 0 1 2 3 4 (c) (d) Figure 6: (a) Single Interior mesa (b) Two interior mesas (c) Double boundary half-mesas, or an inversted single interior mesa (d) Two half-mesas at the boundaries and one interior mesa, or an inverted two-mesa pattern. In all four cases, β0 = 0.1 and ε = 0.15. The instability threshold for D is given above the graph. The case (a) has the biggest− stability range here since l u q f(u, w)= f − + wu u2; g(u, w)=1 uw; q 1. (85) − 0 u + q − − As was done in [16], in the limit q 1, the condition (11) reduces to u+ f + w u u2 du 0 f + w u u2 − 0 0 − ∼ ∼− 0 0 + − + Z0 and we obtain to leading order, u− 0; u 3f ; w 4 f /3 as q 0. ∼ + ∼ 0 0 ∼ 0 → p p 00 2 (in fact, u− = O(q) 1). To leading order, the profile U0 then solves U0 f0 +4 f0/3U0 U0 = 0 for y < 0; with U (0)=0= U 0 (0) and U (y)=0 for y > 0 and U u as y− . We then− obtain 0 0 0 0 → + → −∞p 2 −1/4 1/4 −1/2 √3f0 tanh 3 f0 2 y , y< 0 U0 . ∼ ( 0, y> 0 and g− =1, g 1 4f + ∼ − 0 L l0 = . 4f0 Next we compute the correction l1 to interface position using (17). We have 0 [g(U0(y), w0) g+] −∞ − Z 0 2 −1/4 1/4 −1/2 = 4f0 4f0 tanh 3 f0 2 y −∞ − Z ∞ 2 −1/4 1/4 −1/2 1/4 1/2 3/4 =4f0 sech 3 f0 2 y =3 2 4f0 0 Z 20 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 (a) (b) Figure 7: (a) A stable two-mesa solutions to Belouzov-Zhabotinskii model (5). Parameter values are D = 100, ε = 0.1, , q = 0.001 and f0 = 0.61. Circles show the full numerical solution. The solid line shows the asymptotic approximation as computed in Proposition 1, with the mesa half-length l computed to two orders. The dashed line is the same approximation, except l1 is set to zero. (b) Time evolution in the BZ model. Parameter values are the same as in (a), except for f0 =0.63. Initial conditions were given in the form of a two-mesa asymptotic solution, but shifted to the left by 0.05. The two-mesa equilibrium appears to be unstable, though the instability is very slow and the two-mesa solution persists until about t 1E5. This is in good agreement with the theoretical instability threshold of f 0.612 (see text). ∼ 0 ∼ so that 1/4 1/2 −1/4 l1 =3 2 f0 . Finally, we have − − − U 3f tanh2 3 1/4f 1/42 1/2y 3f 1 4 exp(3 1/4f 1/421/2y) as y 0 ∼ 0 0 ∼ 0 − 0 → −∞ p p so that −1/4 1/4 1/2 C+ =4 3f0; µ+ =3 f0 2 ; u+ u2 3f f = p+ = 0 ; w 2 2 Zu− 1 2µ α = 64 3−3/4f 3/423/2 exp( 2µ l ) exp + l + · 0 − + 1 ε − ε 0 1 2−0.53−0.25 = 64 3−3/4f 3/423/2 exp( 4) exp L · 0 − ε − εf 0.75 0 On the other hand, µ− = O (1/q) µ , so that the critical threshold given by (24) becomes + −0.5 −0.25 − 2 3 − D = C ε (1 4f )2 f 1.75 exp L ; C e430.752 10.5 =0.085942,K 2 (86) K 0 − 0 0 εf 0.75 0 ≡ ≥ 0 To verify this formula numerically, we set D = 100, ε = 0.1, q = 0.001,L = 1 and K = 2. Next we solved (1) for several different values of f0, with initial conditions given by the two-mesa steady state approximation on the interval [ 1, 3], perturbed by a small shift of size 0.1. We found the two-mesa state − was unstable with f0 =0.62 or higher but became stable when we took f0 =0.61 or lower – see Figure 7. 21 On the other hand, the threshold value as predicted by (86) with above parameter values and DK = D is f0 =0.6124. Thus we obtain an excellent agreement between the asymptotic theory and direct numerical simulations. In Figure 7(a), the approximation with and without l1 to the steady state is shown. We remark that it was essential to compute the correction l1 to the mesa width; if we were to set l1 = 0 the constant C =0.085942 in (86) would be replaced by 0.00157, a dramatic difference by a factor of e4 50. 0 ∼ 6 Discussion We have examined in detail the route to instability of the K-mesa pattern of (1) as the diffusion coefficient D is increased. The onset of instability occurs for exponentially large D; it is well known that such solution is unstable for the shadow system case D = , see [12]. We have computed explicit instability thresholds ∞ DK given by Theorem 2; as well as mesa dynamics when D is large. The instability thresholds are closely related to the coarsening phenomenon – see Figure 1 for example. This was previously analysed for the Brusselator in [17], where the authors have guessed at the formula for DK , K> 1 given in Theorem 2 by constructing the so-called asymmetric patterns, and without computing the eigenvalues. The formula for DK appears to correspond precisely to the point at which the asymmetric solution bifurcates from the symmetric branch [17]. This is indeed the case when µ l = + 6 µ−(L l), i.e. O(α ) = O(α−). In particular, this is true when either l or L l is sufficiently small. − + 6 − However the study of asymmetric patterns cannot predict the instability thresholds if µ l = µ−(L l): in + − this case, the formula (24) for DK actually depends on K, unlike the former case; whereas the construction of asymmetric patterns is K independent. So in the case µ+l = µ−(L l), the full spectral analysis is unavoidable. − There are some similarities between the instability thresholds for mesa patterns computed here, and instability thresholds for for Gierer-Meinhardt system computed in [30], [31]. [In [30], a singular pertur- bation and matrix algebra approach was used; in [31] an approach using Evans functions and Floquet exponents was used. Here, we use a combination of both: singular perturbations and Floquet exponents.] To be concrete, consider the “standard” GM system, a = ε2a a + a2/h; 0= Dh h + a2. (87) t xx − xx − Unlike the K mesa patterns considered in this paper, the steady state for GM system considered in [30] consists of K spikes, concentated at K symmetrically spaced points. The authors derived a sequence of thresholds ? 2 D1 = ε exp(2/ε)/125 (88) ? 1 DK = 2 ,K 2 (89) K ln √2+1 ≥ ? ? such that K spikes on the interval [ 1, 1] are stable if D ε 1 2µ−g+ D1 = O exp ; K Kε g g− + − ε 1 2µ+g− DK = O exp if µ+l < µ−(L l), K> 1. K Kε g− g − − + ? Thus DK is exponenetially large in ε for all K, whereas DK is O(1) for K > 1 and exponentially large for K = 1. We remark that the GM model with saturation (7) exhibits mesa patterns when the saturation is sufficiently large, but exhibits spikes when saturation is small. It is an interesting open question to elucidate the transition mechanism whereby a mesa can become a spike and how the various instability thresholds change from being exponentially large to algebraically large as saturation is decreased. 22 The coarsening phenomenon observed in reaction-diffusion systems is also reminiscent of Oswald ripening in thin fluids – see for example [32], [33] and references therein. 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